Different varieties of the tropical flower Heleconia are fertilized by different species of hummingbirds. Over time, the lengths of the flows and the form of hummingbirds’ beaks have evolved to match each other. Here are data on the lengths in millimeters of two color varieties of the same species of flower in the island of Dominica.

Is the enough evidence that the mean lengths of the two varieties differ? Estimate the difference between the population means, with a 95% confidence.

**Solution: **We have the following table:

We need to test the following null and alternative hypotheses:

_{}

which corresponds to a two-tailed independent samples t-test. Before computing the confidence interval, we need to test whether the variances can be considered equal or not. We need to test

_{}

The F-statistics is computed as

_{}

The lower and upper critical values for_{} and *df*_{1} = 22 and *df*_{2} = 14 are

_{}

which means that we reject the null hypothesis. Since we assume that the variances are unequal, the t-statistics is computed as

_{}

The critical values for this two-tailed test are_{}, and the rejection region is therefore

_{}

Since we have that_{} >_{}, then we reject the null hypothesis.

*This means that we have enough evidence to support the claim that the means are different, at the 0.05 significance level.*

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